LECTURE 9 STRESS - STRAIN RELATIONS
A linear elastic material is one in which the strain is proportional to stress as shown below: There are also other types of idealized models of material behavior.
Previously stress – strain relations were considered for the special case of a uniaxial loading i.e. only one component of stress i.e. the axial or normal component of stress was coming into picture. In this section we shall generalize the elastic behavior, so as to arrive at the relations which connect all the six components of stress with the six components of elastic stress. Futher, we would restrict overselves to linearly elastic material. Before writing down the relations let us introduce a term ISOTROPY
Examples of anisotropic materials, whose properties are different in different directions are (i) Wood (ii) Fibre reinforced plastic (iii) Reinforced concrete
These equation expresses the relationship between stress and strain (Hook's law) for uniaxial state of stress only when the stress is not greater than the proportional limit. In order to analyze the deformational effects produced by all the stresses, we shall consider the effects of one axial stress at a time. Since we presumably are dealing with strains of the order of one percent or less. These effects can be superimposed arbitrarily. The figure below shows the general triaxial state of stress.
Let us consider a case when s
Therefore the resulting strains in three directions are Similarly let us consider that normal stress s
Now let us consider the stress s In the following analysis shear stresses were not considered. It can be shown that for an isotropic material's a shear stress will produce only its corresponding shear strain and will not influence the axial strain. Thus, we can write Hook's law for the individual shear strains and shear stresses in the following manner. The Equations (1) through (6) are known as Generalized Hook's law and are the constitutive equations for the linear elastic isotropic materials. When these equations isotropic materials. When these equations are used as written, the strains can be completely determined from known values of the stresses. To engineers the plane stress situation is of much relevance ( i.e. s Hook's law is probably the most well known and widely used constitutive equations for an engineering materials.” However, we can not say that all the engineering materials are linear elastic isotropic ones. Because now in the present times, the new materials are being developed every day. Many useful materials exhibit nonlinear response and are not elastic too.
So if we take the xy plane then s A plane stress may be defined as a stress condition in which all components associated with a given direction ( i.e the z direction in this example ) are zero
Î i.e. if strain components Î
In considering the elastic behavior of an isotropic materials under, normal, shear and hydrostatic loading, we introduce a total of four elastic constants namely E, G, K, and g . It turns out that not all of these are independent to the others. In fact, given any two of them, the other two can be foundout . Let us define these elastic constants (i) E = Young's Modulus of Rigidity
Let us find the relations between them |